#### 3.6 Network Equations

For a network with buses, all constant impedance elements of the model
are incorporated into a complex bus admittance matrix that
relates the complex nodal current injections to the complex node voltages
:

| (3.8) |

Similarly, for a network with branches, the system branch admittance
matrices and relate the bus voltages to the vectors and of branch
currents at the from and to ends of all branches, respectively:

If is used to denote an operator that takes an vector and creates the
corresponding diagonal matrix with the vector elements on the diagonal, these
system admittance matrices can be formed as follows:

The current injections of (3.8)–(3.10) can be used to compute the corresponding
complex power injections as functions of the complex bus voltages :

The nodal bus injections are then matched to the injections from loads and
generators to form the AC nodal power balance equations, expressed as a function
of the complex bus voltages and generator injections in complex matrix form
as

| (3.17) |